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Wednesday, October 10, 2007

M Theory Lesson 111

A commenter at Carl Brannen's blog has noted the similarity between the snuark mass computations and the Fano plane, which we recall describes the octonions via a cube with corners $1, e_1, e_2, \cdots , e_7$. In Carl's notation, the correspondence is

$e_1 = +y$$e_2 = +z$$e_3 = -y$$e_4 = +x$$e_5 = -x$$e_6 = -z$$e_7 = 0$

where $0$ is the fictitious vacuum and the $1$ is hidden at the rear of his diagrams. On the octonion cube, this gives a source of $0$ and a target of $1$, with $x$, $y$ and $z$ axes running through the other three full diagonals. Note that a projection of this cube onto a plane results in a hexagon with vertices $+x, -z, +y, -x, +z, -y$, which has a selected triple of nodes $(x,y,z)$ as previously noted. The basic simplex formed from the source in these directions is the area marked with the phase $\frac{\pi}{24}$ in Carl's computation.

Aside: Note also the new paper by Yidun Wan on 3-braids, which discusses Veneziano bubbles and rotations by $\frac{\pi}{3}$ and $\frac{2 \pi}{3}$, the symmetries of a triangle.